Abstract
Merge trees are fundamental data structures in computational topology. They track connected components in sublevel sets of scalar functions and can be used to compute 0-dimensional persistence diagrams, to construct contour trees on simply connected domains, and to quickly query the relationship between connected components in different sublevel sets. We introduce a representation of merge trees that tracks the nesting of their branches. We present algorithms to construct and manipulate the trees in this representation directly. We show that our algorithms are not only fast, outperforming Kruskal’s algorithm, but they are easy to parallelize in shared memory using double-word compare-and-swap operations. We present experiments that illustrate the scaling of our algorithms as functions of the data size and of the number of threads.
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Notes
- 1.
Given a sequence of n values, take a star graph with n leaves and assign the input values to the leaves; assign −∞ to the central vertex. The merge tree of this function is a path, with vertices assigned the input values in sorted order.
- 2.
We stopped at two data points because by the third, the jobs exhausted the 4-h wallclock request.
- 3.
On x86-64 architecture, where we conducted all our experiments, DWCAS is performed by instruction CMPXCHG16B. It is automatically emitted by the C++ compilers, when provided with appropriate flags.
- 4.
This is important because in C++, once a variable is declared std::atomic, all operations on it are protected by atomic primitives and incur the corresponding overheads.
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Acknowledgements
This work was supported by the Director, Office of Science, Office of Advanced Scientific Computing Research, of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231, through the grant “Scalable Data-Computing Convergence and Scientic Knowledge Discovery,” and by the use of resources of the National Energy Research Scientific Computing Center (NERSC).
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Smirnov, D., Morozov, D. (2020). Triplet Merge Trees. In: Carr, H., Fujishiro, I., Sadlo, F., Takahashi, S. (eds) Topological Methods in Data Analysis and Visualization V. TopoInVis 2017. Mathematics and Visualization. Springer, Cham. https://doi.org/10.1007/978-3-030-43036-8_2
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DOI: https://doi.org/10.1007/978-3-030-43036-8_2
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